GENERATING FUNCTIONS FOR REAL ... - Mathematics

GENERATING FUNCTIONS FOR REAL CHARACTER DEGREE SUMS OF FINITE GENERAL LINEAR AND UNITARY GROUPS JASON FULMAN AND C. RYAN VINROOT Abstract. We compute generating functions for the sum of the realvalued character degrees of the finite general linear and unitary groups, through symmetric function computations. For the finite general linear group, we get a new combinatorial proof that every real-valued character has Frobenius-Schur indicator 1, and we obtain some q-series identities. For the finite unitary group, we expand the generating function in terms of values of Hall-Littlewood functions, and we obtain combinatorial expressions for the character degree sums of real-valued characters with Frobenius-Schur indicator 1 or −1. 2010 AMS Mathematics Subject Classification: 20C33, 05A15

1. Introduction Suppose that G is a finite group, Irr(G) is the set of irreducible complex characters of G, and χ ∈ Irr(G) affords the representation (π, V ). Recall that the Frobenius-Schur indicator of χ (or of π), denoted ε(χ), takes only the values 1, −1, or 0, where ε(χ) = ±1 if and only if χ is real-valued, and ε(χ) = 1 if and only if (π, V )P is a real representation [9, Theorem 4.5]. From the formula ε(χ) = (1/|G|) g∈G χ(g 2 ) [9, Lemma 4.4], it follows that we have [9, Corollary 4.6] X X X (1) ε(χ)χ(1) = χ(1) − χ(1) = #{h ∈ G | h2 = 1}. χ∈Irr(G)

χ∈Irr(G) ε(χ)=1

χ∈Irr(G) ε(χ)=−1

In particular, from (1) it follows that the statement that ε(χ) = 1 for every real-valued χ ∈ Irr(G) is equivalent to the statement that the sum of the degrees of all real-valued χ ∈ Irr(G) is equal to the right side of (1). The main topic of this paper is to study sums of degrees of real-valued characters from a combinatorial point of view of the general linear and unitary groups defined over a finite field Fq with q elements, which we denote by GL(n, q) and U (n, q), respectively. Before addressing the main question of the paper, we begin in Section 2 by examining the classical Weyl groups. In particular, we consider the Weyl Key words and phrases. Frobenius-Schur indicator, real-valued characters, involutions, finite general linear group, finite unitary group, generating functions, q-series, HallLittlewood polynomial. 1

2

JASON FULMAN AND C. RYAN VINROOT

groups of type A (symmetric groups), type B/C (hyperoctahedral groups), and type D. It is well known that every complex irreducible representation of every finite Coxeter group is defined over the real numbers, and moreover, it was proved using a unified method by Springer that all complex representations of Weyl groups are defined over Q [15]. We consider the classical Weyl groups, however, as they serve as natural examples to demonstrate the method of generating functions and symmetric function identities to calculate the (real) character degree sum of a group. In Section 3, we concentrate on the real character degree sum of GL(n, q). It is in fact known that ε(χ) = 1 for every real-valued χ ∈ Irr(GL(n, q)). This was first proved for q odd by Ohmori [12], and it follows for all q by a result of Zelevinsky [20, Proposition 12.6]. So the real character degree sum is known to be the number of elements of GL(n, q) which square to 1. We obtain this result independently by calculating the sum of the degrees of the real-valued characters through symmetric function computations, and applying q-series identities. We also obtain some q-series identities in the process. In Theorem 3.2, we give a generating function for the real character degree sum of GL(n, q) from symmetric function calculations. In Theorem 3.4, we recover Zelevinsky’s result on the Frobenius-Schur indicators of characters of GL(n, q) in the case that q is even, and we record the corresponding q-series identity in Corollary 3.5. We recover Zelevinsky’s result for the case that q is odd in Theorem 3.6, where the calculation is a bit more involved than in the case that q is even. The resulting q-series identity, in Corollary 3.7, seems to be an interesting result in its own right. In Section 4, we turn to the problem of calculating the real character degree sum for the finite unitary group U (n, q). The main motivation here is that given a real-valued χ ∈ Irr(U (n, q)), it is unknown in general whether ε(χ) = 1 or ε(χ) = −1, although the values of ε(χ) are known for certain subsets of characters of U (n, q), such as the unipotent characters [14] and the regular and semisimple characters [16]. Unlike the GL(n, q) case, it is known that there are χ ∈ Irr(U (n, q)) such that ε(χ) = −1. Using symmetric function techniques similar to the GL(n, q) case, we compute a generating function for the real character degree sum of U (n, q) in Theorem 4.1. The point here is that from Equation (1), by counting the number of elements in U (n, q) which square to 1, we have the difference of the real character degree sums of those χ such that ε(χ) = 1, minus those χ such that ε(χ) = −1. Using the generating function in Theorem 4.1 for the sum (rather than the difference) of these character degree sums, along with applying the q-series identities obtained in the GL(n, q) case, we obtain a generating function for the character degree sums of χ ∈ Irr(U (n, q)) satisfying ε(χ) = 1, and another for those satisfying ε(χ) = −1, in Corollaries 4.5 and 4.10. We are then able to expand the generating function obtained in Theorem 4.1, using results of Warnaar [19], stated in Theorem 4.2 and Corollary 4.3. For q even, we give the resulting expression for the sum of the real character degrees of U (n, q) in Theorem 4.6, and for q odd, in Theorem

GENERATING FUNCTIONS FOR REAL CHARACTER DEGREE SUMS

3

4.11. These expressions contain, among other things, special values of HallLittlewood functions of the form Pλ (1, t, t2 , . . . ; −t), where t = −q −1 . Since these values do not seem to be well understood, the expressions we obtain are somewhat complicated in terms of calculation, but we compute several examples to verify the expressions for small n. The fact that these values of Hall-Littlewood functions show up in representation theory gives motivation to better understand them, and the fact that the expressions we obtain are complicated may reflect the fact that it has been a difficult problem to understand the Frobenius-Schur indicators of the characters of U (n, q). We hope that a better understanding of the combinatorial expressions we obtain in Theorems 4.6 and 4.11 will reveal interesting character-theoretic information for the finite unitary groups. 2. Examples: Classical Weyl groups As was mentioned above, it is well understood that every complex irreducible character χ of a finite Coxeter group satisfies ε(χ) = 1. To motivate the methods we will use for the groups GL(n, q) and U (n, q), we consider this fact for the classical Weyl groups, from the perspective of generating functions and Schur function identities. 2.1. Symmetric groups. It is well known (see [5], for example) that the number of elements which square to the identity in the symmetric group Sn (the Weyl group of type An−1 ) is exactly n! times the coefficient of un in 2 eu+u /2 . On the other hand, the irreducible complex representations of Sn are parameterized by partitions λ of n, and if dλ is the degree of the irreducible character χλ labeled by the Q partition λ of n, then d(λ) is given by the hooklength formula, dλ = n!/ b∈λ h(b), where h(b) is the hook length of a box b in the diagram for λ. By [10, I.3, Example 5], we have n! dλ = Q = n! · lim sλ (1/m, . . . , 1/m), m→∞ b∈λ h(b) where sλ (1/m, . . . , 1/m) is the Schur function in m variables, evaluated at 1/m for each variable. The Schur symmetric functions are studied extensively in [10, Chapter I]. Using the identity [10, I.5, Example 4] X Y Y (2) sλ (x) = (1 − xi )−1 (1 − xi xj )−1 , λ

i

i